Q.In a petrol engine (internal combustion engine), air at atmospheric pressure and a temperature of 20°C is adiabatically compressed in the cylinder by the piston to 1/8 of its original volume. Calculate the temperature of the compressed air. (For air, $\gamma = 1.4$.)

An adiabatic process is one in which no heat flows into or out of the system at all ($Q=0$), so the first law reduces to $\Delta U = -W$: the gas can only expand at the expense of its own internal energy (cooling as it does so), or be compressed by external work done on it (heating as a result). This can happen either because the system is thermally insulated from its surroundings, or because the process happens so fast that there simply isn't time for heat to flow even without insulation (e.g. a bursting tyre, or the rapid compression stroke of a diesel engine, or warm air rising and adiabatically expanding and cooling to form clouds). The equation of state for a quasi-static adiabatic process is $PV^\gamma = \text{constant}$, where $\gamma=C_p/C_v$ is the adiabatic exponent; equivalently $TV^{\gamma-1}=\text{constant}$ or $T^\gamma P^{1-\gamma}=\text{con …